
It has been shown in [1]} that the optimal phase shifts should align all IRS-reflected and non-IRS-reflected signals to maximize its effective UL channel power gain, which are given by \([\mathbf {v}^\star _k]_n= e^{\jmath ( \arg \lbrace h^H_{d,k}\rbrace  - \arg \lbrace  [\mathbf {q}^H_k]_n\rbrace  )}, \forall n\) . Define \(\gamma _k \triangleq |h^H_{d,k} + \mathbf {q}^H_k \mathbf {v}^\star _k|^2\)  and \( | {\bar{\mathbf {q}}}^H_k \bar{\mathbf {v}}_0 | \triangleq | h^H_{d,k} + \mathbf {q}^H_k \mathbf {v}_0 | \) , where \({\bar{\mathbf {v}}}_0 = [ \mathbf {v}^H_0 \: 1]^H\)  and \( {\bar{\mathbf {q}}}^H_k = [ {\mathbf {q}}^H_k \: h^H_{d,k}] \) . Then, (P1) can be written as
\(\mathop {\mathrm {max} }\limits _{{\tau _{0},\lbrace \tau _{k}\rbrace ,\lbrace p_{k}\rbrace , {\bar{\mathbf {v}}}_0 } }~~ &\sum _{k=1}^{K} \tau _k\log _2\left(1+ \frac{p_k \gamma _k }{\sigma ^2} \right) \\\mathrm {s.t.} ~~~~~~~& {p_k}\tau _k\le \eta _kP_{\rm A} | {\bar{\mathbf {q}}}^H_k \bar{\mathbf {v}}_0 |^2\tau _0, ~ \forall k,  \\& (\ref {SecII:eq402}), (\ref {SecII:eq403}), |[\bar{\mathbf {v}}_0]_n|=1, n=1,\cdots , N+1. \) 
